Theorem mulgnnp1 13667

Description: Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)

Hypotheses

Ref	Expression
mulg1.b	|- B = ( Base ` G )
mulg1.m	|- .x. = (.g ` G )
mulgnnp1.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
mulgnnp1	|- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) )

Proof of Theorem mulgnnp1

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> N e. NN )
2		nnuz 9758	. . . . 5 |- NN = ( ZZ>= ` 1 )
3	1, 2	eleqtrdi 2322	. . . 4 |- ( ( N e. NN /\ X e. B ) -> N e. ( ZZ>= ` 1 ) )
4		simplr 528	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> X e. B )
5		simpr 110	. . . . . 6 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> u e. ( ZZ>= ` 1 ) )
6	5, 2	eleqtrrdi 2323	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> u e. NN )
7		fvconst2g 5853	. . . . . . 7 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) = X )
8		simpl 109	. . . . . . 7 |- ( ( X e. B /\ u e. NN ) -> X e. B )
9	7, 8	eqeltrd 2306	. . . . . 6 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. B )
10	9	elexd 2813	. . . . 5 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. _V )
11	4, 6, 10	syl2anc 411	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { X } ) ` u ) e. _V )
12		simprl 529	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> u e. _V )
13		mulg1.b	. . . . . . . 8 |- B = ( Base ` G )
14	13	basmex 13092	. . . . . . 7 |- ( X e. B -> G e. _V )
15		mulgnnp1.p	. . . . . . . 8 |- .+ = ( +g ` G )
16		plusgslid 13145	. . . . . . . . 9 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
17	16	slotex 13059	. . . . . . . 8 |- ( G e. _V -> ( +g ` G ) e. _V )
18	15, 17	eqeltrid 2316	. . . . . . 7 |- ( G e. _V -> .+ e. _V )
19	14, 18	syl 14	. . . . . 6 |- ( X e. B -> .+ e. _V )
20	19	ad2antlr 489	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> .+ e. _V )
21		simprr 531	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> v e. _V )
22		ovexg 6035	. . . . 5 |- ( ( u e. _V /\ .+ e. _V /\ v e. _V ) -> ( u .+ v ) e. _V )
23	12, 20, 21, 22	syl3anc 1271	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> ( u .+ v ) e. _V )
24	3, 11, 23	seq3p1 10687	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ ( ( NN X. { X } ) ` ( N + 1 ) ) ) )
25		id 19	. . . . 5 |- ( X e. B -> X e. B )
26		peano2nn 9122	. . . . 5 |- ( N e. NN -> ( N + 1 ) e. NN )
27		fvconst2g 5853	. . . . 5 |- ( ( X e. B /\ ( N + 1 ) e. NN ) -> ( ( NN X. { X } ) ` ( N + 1 ) ) = X )
28	25, 26, 27	syl2anr 290	. . . 4 |- ( ( N e. NN /\ X e. B ) -> ( ( NN X. { X } ) ` ( N + 1 ) ) = X )
29	28	oveq2d 6017	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ ( ( NN X. { X } ) ` ( N + 1 ) ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
30	24, 29	eqtrd 2262	. 2 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
31		mulg1.m	. . . 4 |- .x. = (.g ` G )
32		eqid 2229	. . . 4 |- seq 1 ( .+ , ( NN X. { X } ) ) = seq 1 ( .+ , ( NN X. { X } ) )
33	13, 15, 31, 32	mulgnn 13663	. . 3 |- ( ( ( N + 1 ) e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) )
34	26, 33	sylan 283	. 2 |- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) )
35	13, 15, 31, 32	mulgnn 13663	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) )
36	35	oveq1d 6016	. 2 |- ( ( N e. NN /\ X e. B ) -> ( ( N .x. X ) .+ X ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
37	30, 34, 36	3eqtr4d 2272	1 |- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   {csn 3666   X. cxp 4717   ` cfv 5318  (class class class)co 6001   1c1 8000   + caddc 8002   NNcn 9110   ZZ>=cuz 9722   seqcseq 10669   Basecbs 13032   +g cplusg 13110  ._gcmg 13656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447  df-uz 9723  df-seqfrec 10670  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-minusg 13537  df-mulg 13657

This theorem is referenced by:  mulg2  13668  mulgnn0p1  13670  mulgnnass  13694  gsumfzconst  13878